Disclaimer

Fermions summary

We’ve considered a momentum sphere as in fig. 1.1, and performed various appromations of the occupation sums fig. 1.2.

Fig 1.1: Summation over momentum sphere

Fig 1.2: Fermion occupation

The physics of Fermi gases has an extremely wide range of applicability. Illustrating some of this range, here are some examples of Fermi temperatures (from )

Electrons in copper:

Neutrons in neutron star:

Ultracold atomic gases:

Bosons

We’d like to work with a fixed number of particles, but the calculations are hard, so we move to the grand canonical ensemble

Again, we’ll consider free particles with energy as in fig. 1.3, or

Fig 1.3: Free particle energy momentum distribution

Again introducing fugacity , we have

We’ll consider systems for which

Observe that at large energies we have

For small energies

Observe that we require (or ) so that the number distribution is strictly positive for all energies. This tells us that the fugacity is a function of temperature, but there will be a point at which it must saturate. This is illustrated in fig. 1.4.

Fig 1.4: Density times cubed thermal de Broglie wavelength

Let’s calculate this density (assumed fixed for all temperatures)

With the substitution

we find

This implicitly defines a relationship for the fugacity as a function of temperature .

It can be shown that

As we end up with a zeta function, for which we can look up the value

where the Riemann zeta function is defined as

At high temperatures we have

(as does down, goes up)

Looking at leads to

How do I satisfy number conservation?

We have a problem here since as the term in above drops to zero, yet cannot keep increasing without bounds to compensate and keep the density fixed. The way to deal with this was worked out by

Disclaimer

Fermi gas

Review

Continuing a discussion of [1] section 8.1 content.

We found

With no spin

Fig 1.1: Occupancy at low temperature limit

Fig 1.2: Volume integral over momentum up to Fermi energy limit

gives

This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where

Moving on

with

this gives

Over all dimensions

so that

Again

Example: Spin considerations

{example:basicStatMechLecture16:1}{

This gives us

and again

}

High Temperatures

Now we want to look at the at higher temperature range, where the occupancy may look like fig. 1.3

Fig 1.3: Occupancy at higher temperatures

so that for large we have

Mathematica (or integration by parts) tells us that

so we have

Introducing for the thermal de Broglie wavelength,

we have

Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in fig. 1.4 for a few arbitrarily chosen numerical values of the chemical potential , where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.

Fig 1.4: Density as a function of temperature

We can write

or (taking (and/or volume?) as a constant) we have for large temperatures

The chemical potential is plotted in fig. 1.5, whereas this function is plotted in fig. 1.6. The contributions to from the term are dropped for the high temperature approximation.

Disclaimer

Last time we found that the low temperature behaviour or the chemical potential was quadratic as in fig. 1.1.

Fig 1.1: Fermi gas chemical potential

Specific heat

where

Low temperature

The only change in the distribution fig. 1.2, that is of interest is over the step portion of the distribution, and over this range of interest is approximately constant as in fig. 1.3.

Fig 1.2: Fermi distribution

Fig 1.3: Fermi gas density of states

so that

Here we’ve made a change of variables , so that we have near cancelation of the factor

Here we’ve extended the integration range to since this doesn’t change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have

With , we have for

Using eq. 1.1.4 at the Fermi energy and

we have

Giving

or

This is illustrated in fig. 1.4.

Fig 1.4: Specific heat per Fermion

Relativisitic gas

Relativisitic gas

graphene

massless Dirac Fermion

Fig 1.5: Relativisitic gas energy distribution

We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition. A round trip transition will have to supply energy like as illustrated in fig. 1.6.

Fig 1.6: Hole to electron round trip transition energy requirement

Graphene

Consider graphene, a 2D system. We want to determine the density of states ,

Motivation

I was wondering how to generalize the arguments of [1] to relativistic systems. Here’s a bit of blundering through the non-relativistic arguments of that text, tweaking them slightly.

I’m sure this has all been done before, but was a useful exercise to understand the non-relativistic arguments of Pathria better.

Generalizing from energy to four momentum

Generalizing the arguments of section 1.1.

Instead of considering that the total energy of the system is fixed, it makes sense that we’d have to instead consider the total four-momentum of the system fixed, so if we have particles, we have a total four momentum

where is the total number of particles with four momentum . We can probably expect that the ‘s in this relativistic system will be smaller than those in a non-relativistic system since we have many more states when considering that we can have both specific energies and specific momentum, and the combinatorics of those extra degrees of freedom. However, we’ll still have

Only given a specific observer frame can these these four-momentum components be expressed explicitly, as in

where is the velocity of the particle in that observer frame.

Generalizing the number if microstates, and notion of thermodynamic equilibrium

Generalizing the arguments of section 1.2.

We can still count the number of all possible microstates, but that number, denoted , for a given total energy needs to be parameterized differently. First off, any given volume is observer dependent, so we likely need to map

Let’s still call this , but know that we mean this to be four volume element, bounded in both space and time, referred to a fixed observer’s frame. So, lets write the total number of microstates as

where is the total four momentum of the system. If we have a system subdivided into to two systems in contact as in fig. 1.1, where the two systems have total four momentum and respectively.

Fig 1.1: Two physical systems in thermal contact

In the text the total energy of both systems was written

so we’ll write

so that the total number of microstates of the combined system is now

As before, if denotes an equilibrium value of , then maximizing eq. 1.0.8 requires all the derivatives (no sum over here)

With each of the components of the total four-momentum separately constant, we have , so that we have

as before. However, we now have one such identity for each component of the total four momentum which has been held constant. Let’s now define

Our old scalar temperature is then

but now we have three additional such constants to figure out what to do with. A first start would be figuring out how the Boltzmann probabilities should be generalized.

Equilibrium between a system and a heat reservoir

Generalizing the arguments of section 3.1.

As in the text, let’s consider a very large heat reservoir and a subsystem as in fig. 1.2 that has come to a state of mutual equilibrium. This likely needs to be defined as a state in which the four vector is common, as opposed to just the temperature field being common.

Fig 1.2: A system A immersed in heat reservoir A’

If the four momentum of the heat reservoir is with for the subsystem, and

Writing

for the number of microstates in the reservoir, so that a Taylor expansion of the logarithm around (with sums implied) is

Here we’ve inserted the definition of from eq. 1.0.11, so that at equilibrium, with , we obtain

Next steps

This looks consistent with the outline provided in http://physics.stackexchange.com/a/4950/3621 by Lubos to the stackexchange “is there a relativistic quantum thermodynamics” question. I’m sure it wouldn’t be too hard to find references that explore this, as well as explain why non-relativistic stat mech can be used for photon problems. Further exploration of this should wait until after the studies for this course are done.

Question: Zipper problem ([1] pr 3.7)

A zipper has links; each link has a state in which it is closed with energy and a state in which it is open with energy . we require, however, that the zipper can only unzip from the left end, and that the link number can only open if all links to the left are already open. Find (and sum) the partition function. In the low temperature limit , find the average number of open links. The model is a very simplified model of the unwinding of two-stranded DNA molecules.

Answer

The system is depicted in fig. 1.1, in the and states.

Fig 1.1: Zipper molecule model in first two states

The left opening only constraint simplifies the combinatorics, since this restricts the available energies for the complete molecule to .

The probability of finding the molecule with links open is then

with

We can sum this geometric series immediately

The expectation value for the number of links is

Let’s write

and make a change of variables

so that

The average number of links is thus

or

In the very low temperature limit where (small , big ), we have

showing that on average no links are open at such low temperatures. An exact plot of for a few small values is in fig. 1.2.

Question: Diatomic molecule ([1] pr 4.7)

Consider a classical system of non-interacting, diatomic molecules enclosed in a box of volume at temperature . The Hamiltonian of a single molecule is given by

Study the thermodynamics of this system, including the dependence of the quantity on .

Answer

Partition function
First consider the partition function for a single diatomic pair

Now we can make a change of variables to simplify the exponential. Let’s write

or

Our volume element is

It wasn’t obvious to me that this change of variables preserves the volume element, but a quick Jacobian calculation shows this to be the case

Our remaining integral can now be evaluated

Our partition function is now completely evaluated

As a function of and as in the text, we write

Gibbs sum

Our Gibbs sum, summing over the number of molecules (not atoms), is

or

The fact that we can sum this as an exponential series so nicely looks like it’s one of the main advantages to this grand partition function (Gibbs sum). We can avoid any of the large approximations that we have to use when the number of particles is explicitly fixed.

Pressure

The pressure follows

Average energy

or

Average occupancy

but this is just , or

Free energy

Entropy

Expectation of atomic separation

The momentum portions of the average will just cancel out, leaving just